There is an institute called The International Center in Vina del Mar right next to the Entrance to Quinta Vergara. There are also branches in Santiago, Concepcion and Calama according to their website.
http://internationalcenter.cl/web2009/

I have heard mostly good things about this institute and know people who've worked there. The good things are that schedules often work in shifts of around 3 hours, so sometimes you might have a convenient set schedule where you're not commuting around all day. They have one of the highest institute pays in the area, though it also varies depending on what type of work you do, but I've heard pay can be near 7,000 in some cases. (I've gathered some people must supervise labs or listening or something, some go teach at companies, some do group or private at the institute, and some lucky few get to teach CORFO classes which pay around $10,000 per hour). This might not be exactly the case or true anymore but employees have told me this.

However, I've heard that the situation with taxes is sort of complicated and not always straightforward...they request you to only send boletas (electronic form on the taxes website, chilean freelance version of an invoice style form stating the money requested from employer and the amount retained for taxes) every few months, and many employees have questioned whether they are eligible for the tax money (10%) that corresponds to them.
(I'm not positive THIS school works this way or even lets you work without a permit, but people who work without a visa/permit in chile cannot obviously legally file taxes, however companies sometimes will still remove this money from your paycheck in order to legally account for the work performed, and possibly will attribute the money to someone else, meaning that person will get your money on the tax refund).

I have never worked or applied there so the above is all hearsay, but my impression after working in the area a few years is that this is one of the better institutes to work for. Each branch probably differs in organization and pay might different too, calama and santiago are much more expensive cities, for example.